Milburn, who was Labour health minister under Tony Blair, said he repeatedly approached No 10 and No 11 for support, to no avail.

David Cameron’s had government asked his Commission to report directly to the PM, he added, but discussions “went into the void” when May took power and “got worse” after the General Election.

Milburn told the Commons Education Committee that “it didn’t seem that the government had either the ability or the willingness” to put “shoulders to the wheel,” despite some valiant efforts by Justine Greening - the former Education Secretary effectively sacked in the Cabinet reshuffle.

He went on: “Now, why was that? It’s pretty obvious in a sense and perhaps it’s understandable. There is a huge issue before the country and that is Brexit. It’s fiercely complex. It’s probably the most complex international negotiation that there has ever been.

“The conclusion that I certainly reached was that the Government lacked the headspace or the bandwidth to match the rhetoric of healing social division with the reality.”

Saying Brexit had left the Commission’s work in “no man’s land”, he added: “There was no conversation. There was no response.”

A number of posts with the body lay vacant and Milburn made recommendations but “all were rejected” and “it became increasingly obvious there was disengagement”.

Peter Nicholls / Reuters
Theresa May speaking to the media on the steps of Downing Street

In June, the Commission warned that without radical and urgent reform, the social and economic divisions tearing Britain apart will widen even further and threaten community cohesion and economic prosperity.

It also highlighted that young people in poor towns were offered fewer opportunities, that the rich-poor divide was widening and intergenerational unfairness was a growing concern.

Milburn told MPs May had failed to deliver on the promises made in her first speech on the steps of Downing Street.

“It [social mobility] is above party politics and it is the crunch issue for our country,” he said. “There is a social crisis in the country.”

But he added: “For the avoidance of doubt, I don’t for a minute question the PM’s commitment to social justice. (..)

“My question is less about the commitment and more about the ability to deliver.”

Conservative MP Lucy Allan observed that “the commission was allowed to dwindle on the vine” and added “perhaps that was an intentional act”.

Shephard added: “We had to infer that the Government wanted to move on.”

PA Archive/PA Images
Tory MP Lucy Allan suggested that the Social Mobility Commission was "allowed to wither on the vine"

Milburn said the issue required deep thinking, and said: “You need an effort for the long-term and you need a strategy. There isn’t a strategy. Let’s be honest about it.”

He feared a generation would lose hope.

“It [social mobility] is something that affects every ordinary low and middle income member of our society,” he said. “It’s about the prospects of their kids.

“I was brought up to believe a very simple thing, that if I put the effort in I would get a reward. I was very lucky in my life. I grew up on a council estate and I ended up in the Cabinet.

“What I worry is that isn’t possible anymore and a lot of people believe that to be the case, and it’s wrong. It’s wrong.”

The most pressing issue for May was reform of the labour market to rebalance the economy away from London, Milburn said, with “middle class brain drain” affecting the North and the Midlands and him claiming that “once people leave, they never go back”.

He said that while 700,000 professional jobs were created in London, just 140,000 were in the Midlands and in the North East the figure was less than 60,000

“We can’t keep doing this,” he said. “It doesn’t work for anyone. It doesn’t work for housing prices in London any more than it works for the lack of job opportunities in the North East of England.

“That requires government – government – to step in and right now, we’re in a no man’s land.”